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9 Summary of Algorithms/Codes for Finding Apparent Horizons

There are many apparent horizon finding algorithms, with differing trade-offs between speed, robustness of convergence, accuracy, and ease of programming.

In spherical symmetry, zero-finding (Section 8.1) is fast, robust, and easy to program. In axisymmetry, shooting algorithms (Section 8.2) work well and are fairly easy to program. Alternatively, any of the algorithms for generic slices (summarized below) can be used with implementations tailored to the axisymmetry.

Minimization algorithms (Section 8.3) are fairly easy to program, but when the underlying simulation uses finite differencing these algorithms are susceptible to spurious local minima, have relatively poor accuracy, and tend to be very slow unless axisymmetry is assumed. When the underlying simulation uses spectral methods, then minimization algorithms can be somewhat faster and more robust.

Spectral integral-iteration algorithms (Section 8.4) and elliptic-PDE algorithms (Section 8.5) are fast and accurate, but are moderately difficult to program. Their main disadvantage is the need for a fairly good initial guess for the horizon position/shape.

In many cases Schnetter’s “pretracking” algorithm (Section 8.6) can greatly improve an elliptic-PDE algorithm’s robustness, by determining – before it appears – approximately where (in space) and when (in time) a new outermost apparent horizon will appear. Pretracking is implemented as a modification of an existing elliptic-PDE algorithm and is moderately slow: It typically has a cost 5 to 10 times that of finding a single horizon with the elliptic-PDE algorithm.

Finally, flow algorithms (Section 8.7) are generally quite slow (Metzger’s algorithm [109] is a notable exception) but can be very robust in their convergence. They are moderately easy to program. Flow algorithms are global algorithms, in that their convergence does not depend on having a good initial guess.

Table 2 lists freely-available apparent horizon finding codes.

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